Adding a keyframed Brownian Motion to an object
You could add keyframe modifiers.


Add keyframe to the x,y,z positionsn at the beginning of the animation
Go to the Graph Editor -> right panel (press N to show it) -> "Modifiers" label -> Add Modifier -> Noise.
Do the operation 2 to the three keyframes (X, Y and Z positions).
In the Modifier panel (where you just added the modifier) of each keyframe (X, Y and Z) modify the noise properties in order to make them different each other and to set the Brownian motion as you want.